The toilet often keeps running until you wiggle the flush handle.
This symptom signals an adjustment problem, either in the flapper chain or tank-ball linkage. If your toilet has a flapper, the lift chain may be too long. To correct the problem, remove the tank lid, and lift the chain from the wire hook that secures it has less slack. With the chain hooked, top and bottom, press it to one side. You should see roughly 1 inch of sideways deflection.
If your toilet has a tank ball, expect the liftwire guide to be out of alignment. You'll find this guide clamped around the flush valve's overflow tube, secured by a setscrew. Begin by removing the tank lid; then flush the toilet several times. Keep flushing until you see the tank ball fall off-center, showing you which way to move the guide. If the ball falls to the left, for example, move the guide to the right.
Shut off the water, and loosen the setscrew about two full turns. Rotate the guide about 1/16inch, and then reset the screw. Turn the water back on, and watch tank ball fall through several more flushes. If you've under-or over-corrected, you should be able to see it in the way the ball hits the flushvavle seat. It should hit dead center when it lands.
When making these adjustments, try not apply too much pressure to the brass overflow tube. Brass gets brittle with age, and an old overflow tube might break off. If this should happen, simply pry the remaining bit of tube from its valve threads, and screw a new brass tube in place.